Are there polyominoes which tile half-plane but tile no strip with any width?

Question

In Golomb's hierarchy: If a polyomino tiles strip then tiles half-plane. (Ok, it's trivial.) But what is with other direction? Is there an example which tiles half-plane but doesn't tile any strip?

Answer 1

(Not a solution)
OP clarified that they want "example tile which tiles half-plane but doesn't tile any strip".

This solution interprets the question as "example tiling which tiles half-plane but doesn't tile any strip".

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Prove by induction that the $L-$polynomino consisting of 3 squares can perfectly tile a $2^n \times 2^n$ grid with 1 corner square removed such that every internal grid line intersects with a tile.

Use this to generate a tiling of the quarter-plane and reflect it to get the half-plane, but there is no strip since the only grid line that doesn't intersect a tile is $ y = 0 $.